Chapter 3.8, Problem 20P

Solve each inequality. Write the solutions as either the union or intersection of two sets.

$|2x-3|\le 11$

To determine

To solve the inequality $\left|2x-3\right|\ge 11$ and write the solution as either union or intersection of two sets.

Answer to Problem 20P

The solution of inequality $\left|2x-3\right|\ge 11$ is $\left(-\infty ,-4\right]\cup \left[7,\infty \right)$

Explanation of Solution

Given information :

Given inequality $\left|2x-3\right|\ge 11$

Given inequality $\left|2x-3\right|\ge 11$

  $\begin{array}{l}\Rightarrow 2x-3\le -11\text{or}2x-3\ge 11\\ \Rightarrow 2x\le -11+3\text{or}2x\ge 11+3\\ \Rightarrow 2x\le -8\text{or}2x\ge 14\\ \Rightarrow x\le -4\text{or}x\ge 7\\ \therefore x\in \left(-\infty ,-4\right]\cup \left[7,\infty \right)\end{array}$

Thus, we expressed the solution union of two sets.

Hence, the solution of inequality $\left|2x-3\right|\ge 11$ is $\left(-\infty ,-4\right]\cup \left[7,\infty \right)$